Problem: We know that $ 0\le\dfrac{|\cos(n)|}{n}\le \dfrac{1}{n}$ for any $n\ge 1$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=1}^{\infty }\dfrac{|\cos(n)|}{n}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Answer: $\sum_{n=1}^{\infty }~{\frac{1}{{n}}}$ is the harmonic series which is known to diverge. Because our given series is term-by-term less than a divergent series, the direct comparison test does not apply. So the direct comparison test is inconclusive.